The nonlinear radial oscillations of bubbles that are encapsulated in an elastic shell are investigated numerically subject to three different constitutive laws describing the viscoelastic properties of the shell: the Mooney-Rivlin (MR), the Skalak (SK), and the Kelvin-Voigt (KV) models are used in order to describe strain-softening, strain-hardening and small displacement (Hookean) behavior of the shell material, respectively. Due to the isotropic nature of the acoustic disturbances, the area dilatation modulus is the important parameter. When the membrane is strain softening (MR) the resonance frequency decreases with increasing sound amplitude, whereas the opposite happens when the membrane is strain hardening (SK). As the amplitude of the acoustic disturbance increases the total scattering cross section of a microbubble with a SK membrane tends to decrease, whereas that of a KV or a MR membrane tends to increase. The importance of strain-softening behavior in the abrupt onset of volume pulsations, that is often observed with small insonated microbubbles at moderately large sound amplitudes, is discussed.
Stability analysis of the radial pulsations of a gas microbubble that is encapsulated by a thin viscoelastic shell and surrounded by an ideal incompressible liquid is carried out. Small axisymmetric disturbances in the microbubble shape are imposed and their long and short term stability is examined depending on the initial bubble radius, the shell properties, and the parameters, i.e., frequency and amplitude, of the external acoustic excitation. Owing to the anisotropy of the membrane that is forming the encapsulating shell, two different types of elastic energy are accounted for, namely, the membrane and bending energy per unit of initial area. They are used to describe the tensions that develop on the shell due to shell stretching and bending, respectively. In addition, two different constitutive laws are used in order to relate the tensions that develop on the membrane as a result of stretching, i.e., the Mooney-Rivlin law describing materials that soften as deformation increases and the Skalak law describing materials that harden as deformation increases. The limit for static buckling is obtained when the external overpressure exerted upon the membrane surpasses a critical value that depends on the membrane bending resistance. The stability equations describing the evolution of axisymmetric disturbances, in the presence of an external acoustic field, reveal that static buckling becomes relevant when the forcing frequency is much smaller than the resonance frequency of the microbubble, corresponding to the case of slow compression. The resonance frequencies for shape oscillations of the microbubble are also obtained as a function of the shell parameters. Floquet analysis shows that parametric instability, similar to the case of an oscillating free bubble, is possible for the case of a pulsating encapsulated microbubble leading to shape oscillations as a result of subharmonic or harmonic resonance. These effects take place for acoustic amplitude values that lie above a certain threshold but below those required for static buckling to occur. They are quite useful in providing estimates for the shell elasticity and bending resistance based on a frequency/amplitude sweep that monitors the onset of shape oscillations when the forcing frequency resonates with the radial pulsation, f = 0 , or with a certain shape mode, f =2 n . An acceleration based instability, identified herein as dynamic buckling, is observed during the compression phase of the pulsation, evolving over a small number of periods of the forcing, when the amplitude of the acoustic excitation is further increased. It corresponds to the RayleighTaylor instability observed for free bubbles, and has been observed with contrast agents as well, e.g., BR-14. Finally, phase diagrams for contrast agent BR-14 are constructed and juxtaposed with available experimental data, illustrating the relevance and range of the above instabilities.
The details of nonlinear oscillations and collapse of elongated bubbles, subject to large internal overpressure, are studied by a boundary integral method. Weak viscous effects on the liquid side are accounted for by integrating the equations of motion across the boundary layer that is formed adjacent to the interface. For relatively large bubbles with initial radius R 0 on the order of millimeters, P St = P St Ј / ͑2 / R 0 ͒ϳ300 and Oh= / ͑R 0 ͒ 1/2 ϳ 200, and an almost spherical initial shape, S ϳ 1, Rayleigh-Taylor instability prevails and the bubble breaks up as a result of growth of higher modes and the development of regions of very small radius of curvature; , , , and P St Ј denote the surface tension, density, viscosity, and dimensional static pressure in the host liquid while S is the ratio between the length of the minor semiaxis of the bubble, taken as an axisymmetric ellipsoid, and its equivalent radius R 0 . For finite initial elongations, 0.5ഛ S Ͻ 1, the bubble collapses either via two jets that counterpropagate along the axis of symmetry and eventually coalesce at the equatorial plane, or in the form of a sink flow approaching the center of the bubble along the equatorial plane. This pattern persists for the above range of initial elongations examined and large internal overpressure amplitudes, B ജ 1, irrespective of Oh. It is largely due to the phase in the growth of the second Legendre mode during the after-bounce of the oscillating bubble, during which it acquires large enough positive accelerations for collapse to take place. For smaller bubbles with initial radius on the order of micrometers, P St ϳ 4 and Ohϳ 20, and small initial elongations, 0.75Ͻ S ഛ 1, viscosity counteracts P 2 growth and subsequent jet motion, thus giving rise to a critical value of Oh −1 below which the bubble eventually returns to its equilibrium spherical shape, whereas above it collapse via jet impact or sink flow is obtained. For moderate elongations, 0.5ഛ S ഛ 0.75, and large overpressures, B ജ 0.2, jet propagation and impact along the axis of symmetry prevails irrespective of Oh. For very large elongations, S Ͻ 0.5, and above a certain threshold value of Oh the counterpropagating jets pinch the contracting bubble sidewalls in an off-centered fashion.
The weak viscous oscillations of a bubble are examined, in response to an elongation that perturbs the initial spherical shape at equilibrium. The flow field in the surrounding liquid is split in a rotational and an irrotational part. The latter satisfies the Laplacian and can be obtained via an integral equation. A hybrid boundary-finite element method is used in order to solve for the velocity potential and shape deformation of axisymmetric bubbles. Weak viscous effects are included in the computations by retaining first-order viscous terms in the normal stress boundary condition and satisfying the tangential stress balance. An extensive set of simulations was carried out until the bubble either returned to its initial spherical shape, or broke up. For a relatively small initial elongation the bubble returned to its initial spherical state regardless of the size of the Ohnesorge number; Oh= / ͑R͒ 1/2 . For larger initial elongations there is a threshold value in Oh −1 above which the bubble eventually breaks up giving rise to a "donut" shaped larger bubble and a tiny satellite bubble occupying the region near the center of the original bubble. The latter is formed as the round ends of the liquid jets that approach each other from opposite sides along the axis of symmetry, coalesce. The size of the satellite bubble decreased as the initial elongation or Oh −1 increased. This pattern persisted for a range of large initial deformations with a decreasing threshold value of the Oh −1 as the initial deformation increased. As its equilibrium radius increases the bubble becomes more susceptible to the above collapse mode. The effect of initial bubble overpressure was also examined and it was seen that small initial overpressures, for the range of initial bubble deformations that was investigated, translate the threshold of Oh −1 to larger values while at the same time increasing the size of the satellite bubble.
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